Their states are still recovering from the recession, and now the nation’s governors are bracing, again, for cuts in federal aid.

They have been down this road before — Congress has already missed several self-imposed deadlines to cut the deficit — but many say they fear that this time, the talks in Washington to avert the so-called fiscal cliff will actually lead to deep cuts.

So they want a say in the negotiations.

“The main message is that it’s important to remember that, on a lot of areas of governance, we’re partners — and that these issues can’t be solved simply by cost-shifting to the states, because the states aren’t really in a position to do all that,” Gov. Jack Markell of Delaware, the chairman of the National Governors Association, said in an interview. “We just want to make sure that we have a voice as these decisions are being made.”

But the federal government has a long history of giving short shrift to the needs of states and cities — by making cuts in federal aid that forced service cuts or tax increases at the local level, or by passing laws requiring localities to take expensive actions without giving them the money to do so.

So in recent days, more than a dozen mayors with the U.S. Conference of Mayors have gone to Washington to lobby lawmakers. And last Monday, Markell, a Democrat, joined several governors from both parties to discuss the issue on a conference call with Vice President Joe Biden.

The states, whose tax collections are still below the peak levels they reached in 2008, are in something of an unusual situation. That is because the automatic tax increases and spending cuts that are scheduled to begin in January, called “the fiscal cliff” by Federal Reserve Chairman Ben Bernanke, are actually better for them in some respects than many of the alternate proposals being discussed in Washington.

Half of the cuts scheduled to take effect at the beginning of next year would be to military spending, which would affect states only indirectly. The scheduled cuts to domestic programs would leave Medicaid, the single biggest source of federal aid to states, untouched. And the planned federal tax increases would increase revenues in states whose tax codes are closely linked to the federal code.

But governors said that no one was rooting for President Barack Obama and Republicans in Congress to fail to reach a financial accord, in part because they fear that the resulting combination of spending cuts and tax increases could prompt another recession, which their states can ill afford.

Gov. Rick Snyder of Michigan, a Republican, noted that the spending cuts and tax increases were intended to be so undesirable that they would spur opposing sides in Washington to overcome their antagonism and strike a deal on taxes and spending just to avoid them.

The plan “was designed to be a terrible answer,” Snyder said, “and I think they did a fairly effective job of doing that.”

Pat McCrory, the Republican governor-elect of North Carolina and former mayor of Charlotte, said state and local officials needed a greater voice in Washington.

“I don’t think the debate should be just between the White House and Capitol Hill, but the state and local government should be at the table,” McCrory said. “Because I assume some of their answers are going to be pass-throughs to the states or to cities, as I saw as mayor in the past.”

The automatic cuts would hurt states in several areas. A recent analysis by the Pew Center on the States found that roughly 18 percent of the federal grant dollars flowing to the states would be subject to across-the-board cuts, including money for education, public housing and nutrition programs for low-income women and children.

But some governors fear that any “grand bargain” struck by Obama and Congress could lead to even deeper cuts to states, and they worry that it could include tax provisions that they believe would be harmful, like ending the tax-exempt status of municipal bonds that makes the bonds more attractive to investors.

But the needs of states and cities have often been an afterthought when Washington has talked about curbing spending.

Alice M. Rivlin, a former director of the Office of Management and Budget who served on two recent high-profile federal commissions — the National Commission on Fiscal Responsibility and Reform, better known as the Simpson-Bowles Commission, and the Bipartisan Policy Center’s Debt Reduction Task Force — acknowledged as much in an appearance over the summer.

“I’ve served on not one but two commissions on the federal deficit,” Rivlin said when another group she belongs to, the State Budget Crisis Task Force, released a report warning of the fiscal problems facing states. “And I can attest that although we were certainly aware that the proposals we made would impact state and local government, we did not do a serious analysis of what would happen.”

Gov. John Kasich of Ohio, a Republican, has been on both sides of the federal-state divide: trying to cut the federal budget as a former chairman of the House Budget Committee, and now seeking to preserve services, especially for the poor, as a governor.

“I’m just saying that if you’re going to affect us, you’d better realize there’s a bottom line that affects flesh and blood and real people,” Kasich told reporters this month at a meeting of the Republican Governors Association in Las Vegas. “And you can easily throw every one of these budgets into the red by just trying to get a nice number.”

The absence of a formal dialogue between the federal government and the states was cited as a danger by the State Budget Crisis Task Force, a private group led by Richard Ravitch, a former lieutenant governor of New York, and Paul A. Volcker, a former Fed chairman.

“There are no standing structures and procedures within the federal government for analyzing the impacts on states and localities of reduced federal spending or federal tax changes, and there is little dialogue about these issues between the federal government and state and local governments,” its report said last summer. It recommended creating something like the Advisory Commission on Intergovernmental Relations, which lasted from 1959 to 1996.

John Kincaid, a former executive director of the advisory commission, which included a bipartisan group of federal and state officials, said it grew less effective as political polarization increased.

“No matter what happens in Washington, it’s going to hit state and local governments very hard,” said Kincaid, who is now a professor of government and public service at Lafayette College in Easton, Pa. “I think, by and large, states have become accustomed to this pattern of decision-making, so they’re going to brace themselves for whatever comes. State and local officials will lobby very hard to get what they can out of this bargain, but they won’t be all that significant as players.”